How can I create Textured Outlines?
Well, you have a few options. 

Freestyle creates a line around the edge of all your objects soon render. You may be able to tweak the settings to create the chalk like effect.

Grease pencil is also another option. Can be harder to accomplish but you have more control over the lines. 

The last option I have is to use a flat plane and tweak texture settings with clouds to create a brush like stroke. Again, difficult and a hassle but could produce some great effects.

If you want any help on how these may be completed I will try and help where I can.

Torus Test:



EDIT:

This is how I created the torus effect above as requested.

Firstly, create a copy of your item slightly larger than your original. The gap between the two will determine the size of your outline.

Create a circle with the triangle fan fill. The number of vertices is your choice, the higher the count, the better the effect will be but performance will be reduced later. I used 128 for mine.

Go to edit mode.

Select the edge ring and delete all the edges, leaving lines spraying from the centre vertex.

Select everything and extrude to create a pretty cool looking mesh.

Select the centre edge (Done by selecting the two centre vertices) and go to Edge-->Edge Split.



Go back to Object mode

This is where thing may get slow for a bit!!

Add a Solidify modifier and set the thickness really low. I set mine to 0.001. 

Add a boolean modifier set to intersect. Choose your slightly larger, copied object for the boolean modifier object. Hide your copy object

Now you should see lots of thin lines around the outside of your mesh as shown below. note as they rotate around they get thinner and disappear.



On my torus, I did not need to do this step for this rotation, but if the centre vertex is showing all clustered together on your meshes surface like below, add a subsurf modifier to the top of your modifiers and set it to 1. this should leave a bit of a hole where the vertex was. BUT DONT WORRY!!! we will fix that soon.

More LAG Ahead!!!

Copy your outline object and duplicate it. Rotate it 90˚ on one of the other two axis' so the lines face in a different direction. Note the small hole on the righthand side due to the subsurf modifier. This will be better hidden with the textures. I do have a small glitch (The plane protruding from my mesh) but it is almost unrecognisable in the render. plus you can delete it once modifiers are applied.



Hide both of these outlines for now if lag is getting the better of your machine :D 

Time for a material. I have taken an image of mine below. if you wanted you could use a texture, but I have gone for a procedural noise texture.



Show your two items again and apply the material. Take a look around in render view. My nodes are set for cycles, so some tweaking may be needed to use in EEVEE.

If you are happy with the results, apply your modifiers IN ORDER!!! and you will get your results in mesh!!

This works by the layering of faces adding their translucency to each other slowly darkening the view. The more layers overlapping, the darker the edge. This does create a little bit of darkening towards the center but when set low like mine it is relitively unoticable.

TADA!

Note that I did this on an early 2015 MacBook air which has relatively low computing power compared to many 3D designers laptops and common desktop PC's. My computer started running quite slow when using the boolean modifier caping my limitations but you may be able to take the number of verticies up in your model.

Good luck with your project and I hope I helped.

BFB